@@ -1,394 +1,394 @@
-\section{Numeric Package}
-\begin{Introduction}{Numeric Package}
-The numeric package supplies algorithms based on approximation
-techniques of numerical mathematics. The algorithms use
-the \nameref{rounded} mode arithmetic of REDUCE, including
-the variable precision feature which is exploited in some
-algorithms in an adaptive manner in order to reach the
-desired accuracy.
-\end{Introduction}
-
-\begin{Type}{Interval} 
-Intervals are generally coded as lower bound and
-upper bound connected by the operator \name{..}, usually 
-associated to a variable in an 
-equation. 
-
-\begin{Syntax}
-  \meta{var} = \(\meta{low} .. \meta{high}\)
-\end{Syntax}
-
-where \meta{var} is a \nameref{kernel} and \meta{low}, \meta{high} are
-numbers or expression which evaluate to numbers with \meta{low}<=\meta{high}.
-
-\begin{Examples}
-     x= (2.5 .. 3.5) 
-\end{Examples}
-
-means that the variable x is taken in the range from 2.5 up to
-3.5. 
-
-
-\end{Type}
-
-\begin{Concept}{numeric accuracy} 
-The keyword parameters \name{accuracy=a} and \name{iterations=i}, 
-where \name{a}and \name{i} must be positive integer numbers, control the
-iterative algorithms: the iteration is continued until
-the local error is below 10**{-a}; if that is impossible
-within \name{i} steps, the iteration is terminated with an
-error message. The values reached so far are then returned
-as the result.
-\end{Concept}
-
-\begin{Switch}{TRNUMERIC}
-Normally the algorithms produce only a minimum of printed
-output during their operation. In cases of an unsuccessful 
-or unexpected long operation a \name{trace of the iteration} can be
-printed by setting \name{trnumeric} \name{on}.
-\end{Switch}
-
-\begin{Operator}{num_min}
-\index{minimum}\index{steepest descent}\index{Fletcher Reeves} 
-The Fletcher Reeves version of the \name{steepest descent}
-algorithms is used to find the \name{minimum} of a
-function of one or more variables. The
-function must have continuous partial derivatives with respect to all
-variables. The starting point of the search can be
-specified; if not, random values are taken instead.
-The steepest descent algorithms in general find only local
-minima.
- 
-\begin{Syntax}
-
- \name{num\_min}\(\meta{exp}, 
-     \meta{var}[=\meta{val}] [,\meta{var}[=\meta{val}] ...
-             [,accuracy=\meta{a}] [,iterations=\meta{i}]\) 
-
-
-or
-
- \name{num\_min}\(exp, \{ 
-      \meta{var}[=\meta{val}] [,\meta{var}[=\meta{val}] ...] \}
-             [,accuracy=\meta{a}] [,iterations=\meta{i}]\) 
-\end{Syntax}
-
-where \meta{exp} is a function expression,
-\meta{var} are the variables in \meta{exp} and
-\meta{val} are the (optional) start values.
-For \meta{a} and \meta{i} see \nameref{numeric accuracy}.
-
-\name{Num\_min} tries to find the next local minimum along the descending
-path starting at the given point. The result is a \nameref{list}
-with the minimum function value as first element followed by a list
-of \nameref{equation}\name{s}, where the variables are equated to the coordinates 
-of the result point.
-
-
-\begin{Examples}
-num_min(sin(x)+x/5, x)&\{4.9489585606,\{X=29.643767785\}\}\\
-num_min(sin(x)+x/5, x=0)&\{ - 1.3342267466,\{X= - 1.7721582671\}\}
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{num_solve}
-\index{equation solving}\index{equation system}\index{Newton iteration}
-\index{root}
-An adaptively damped Newton iteration is used to find
-an approximative root of a function (function vector) or the 
-solution of an \nameref{equation} (equation system). The expressions
-must have continuous derivatives for all variables.
-A starting point for the iteration can be given. If not given
-random values are taken instead. When the number of
-forms is not equal to the number of variables, the
-Newton method cannot be applied. Then the minimum
-of the sum of absolute squares is located instead. 
-
-With \nameref{complex} on, solutions with imaginary parts can be
-found, if either the expression(s) or the starting point
-contain a nonzero imaginary part. 
-
-\begin{Syntax}
- \name{num\_solve}\(\meta{exp}, \meta{var}[=\meta{val}][,accuracy=\meta{a}][,iterations=\meta{i}]\)
-
-or
-
- \name{num\_solve}\(\{\meta{exp},...,\meta{exp}\}, \meta{var}[=\meta{val}],...,\meta{var}[=\meta{val}]
-                [,accuracy=\meta{a}][,iterations=\meta{i}]\)
-
-or
-
- \name{num\_solve}\(\{\meta{exp},...,\meta{exp}\}, \{\meta{var}[=\meta{val}],...,\meta{var}[=\meta{val}]\}
-                [,accuracy=\meta{a}][,iterations=\meta{i}]\)
- 
-\end{Syntax} 
-
-
-where \meta{exp} are function expressions,
-      \meta{var} are the variables,
-      \meta{val} are optional start values.
-For \meta{a} and \meta{i} see \nameref{numeric accuracy}.
- 
-\name{num_solve} tries to find a zero/solution of the expression(s).
-Result is a list of equations, where the variables are
-equated to the coordinates of the result point.
- 
-The \nameindex{Jacobian matrix} is stored as side effect the shared
-variable \name{jacobian}.
-
- 
-\begin{Examples}
-num_solve({sin x=cos y, x + y = 1},{x=1,y=2});&
-   \{X= - 1.8561957251,Y=2.856195584\}\\
-jacobian;&
-\begin{multilineoutput}{5cm}
-    [COS(X)  SIN(Y)]
-    [              ]
-    [  1       1   ]
-\end{multilineoutput}
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{num_int}
-\index{integration} 
-For the numerical evaluation of univariate integrals 
-over a finite interval the following strategy is used:
-If \nameref{int} finds a formal antiderivative
-    which is bounded in the integration interval, this
-    is evaluated and the end points and the difference
-    is returned.
-Otherwise a \nameref{Chebyshev fit} is computed, 
-    starting with order 20, eventually up to order 80.
-    If that is recognized as sufficiently convergent
-    it is used for computing the integral by directly
-    integrating the coefficient sequence.
-If none of these methods is successful, an
-    adaptive multilevel quadrature algorithm is used.
-
-For multivariate integrals only the adaptive quadrature is used.
-This algorithm tolerates isolated singularities. 
-The value \name{iterations} here limits the number of
-local interval intersection levels. 
-\meta{a} is a measure for the relative total discretization
-error (comparison of order 1 and order 2 approximations).
-
- 
-\begin{Syntax}
- \name{num_int}\(\meta{exp},\meta{var}=\(\meta{l} .. \meta{u}\)
-              [,\meta{var}=\(\meta{l} .. \meta{u}\),...]
-             [,accuracy=\meta{a}][,iterations=\meta{i}]\)
-\end{Syntax}
-
-where \meta{exp} is the function to be integrated,
-\meta{var} are the integration variables,
-\meta{l} are the lower bounds,
-\meta{u} are the upper bounds.
- 
-Result is the value of the integral.
- 
-\begin{Examples}
-num_int(sin x,x=(0 .. 3.1415926));& 2.0000010334
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{num_odesolve}
-\index{Runge-Kutta}\index{initial value problem}\index{ODE}
-The \name{Runge-Kutta} method of order 3 finds an approximate graph for
-the solution of real \name{ODE initial value problem}. 
- 
-\begin{Syntax}
- \name{num_odesolve}\(\meta{exp},\meta{depvar}=\meta{start},
-               \meta{indep}=\(\meta{from} .. \meta{to}\)
-                   [,accuracy=\meta{a}][,iterations=\meta{i}]\)
-
-or
-
-\name{num_odesolve}\(\{\meta{exp},\meta{exp},...\},
-                     \{ \meta{depvar}=\meta{start},\meta{depvar}=\meta{start},...\}
-               \meta{indep}=\(\meta{from} .. \meta{to}\)
-                   [,accuracy=\meta{a}][,iterations=\meta{i}]\)
-
-\end{Syntax}
-
-where 
-\meta{depvar} and \meta{start} specify the dependent variable(s)
-and the starting point value (vector), 
-\meta{indep}, \meta{from} and \meta{to} specify the independent variable
-and the integration interval (starting point and end point), 
-\meta{exp} are equations or expressions which
-contain the first derivative of the independent variable
-with respect to the dependent variable.
- 
-The ODEs are converted to an explicit form, which then is
-used for a Runge Kutta iteration over the given range. The 
-number of steps is controlled by the value of \meta{i}
-(default: 20).  If the steps are too coarse to reach the desired 
-accuracy in the neighborhood of the starting point, the number is
-increased automatically.
- 
-Result is a list of pairs, each representing a point of the
-approximate solution of the ODE problem.
-
-\begin{Examples}
-depend(y,x);\\
-num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5);&
-\begin{multilineoutput} 
- \{\{0.0,1.0\},\{0.2,1.2214\},\{0.4,1.49181796\},\{0.6,1.8221064563\},
-  \{0.8,2.2255208258\},\{1.0,2.7182511366\}\}\\
-\end{multilineoutput}
-\end{Examples}
-
-In most cases you must declare the dependency relation
-between the variables explicitly using \nameref{depend};
-otherwise the formal derivative might be converted to zero.
-
-The operator \nameref{solve} is used to convert the form into
-an explicit ODE. If that process fails or if it has no unique result,
-the evaluation is stopped with an error message.
-
-\end{Operator}
-
-\begin{Operator}{bounds}
-
-Upper and lower bounds of a real valued function over an
-\nameref{interval} or a rectangular multivariate domain are computed
-by the operator \name{bounds}. The algorithmic basis is the computation
-with inequalities: starting from the interval(s) of the
-variables, the bounds are propagated in the expression
-using the rules for inequality computation. Some knowledge
-about the behavior of special functions like ABS, SIN, COS, EXP, LOG,
-fractional exponentials etc. is integrated and can be evaluated
-if the operator \name{bounds} is called with rounded mode on 
-(otherwise only algebraic evaluation rules are available).
- 
-If \name{bounds} finds a singularity within an interval, the evaluation
-is stopped with an error message indicating the problem part
-of the expression.
- 
-\begin{Syntax}
-  \name{bounds}\(\meta{exp},\meta{var}=\(\meta{l} .. \meta{u}\) 
-                 [,\meta{var}=\(\meta{l} .. \meta{u}\) ...]\)
-
-or
-
-  \name{bounds}\(\meta{exp},\{\meta{var}=\(\meta{l} .. \meta{u}\) 
-                 [,\meta{var}=\(\meta{l} .. \meta{u}\) ...]\}\)
-
- 
-\end{Syntax}
-
-where \meta{exp} is the function to be investigated,
-\meta{var} are the variables of \meta{exp},
-\meta{l}  and  \meta{u} specify the area as set of \nameref{interval}\name{s}.
-
-\name{bounds} computes upper and lower bounds for the expression in the
-given area. An \nameref{interval} is returned.
- 
-\begin{Examples}
-bounds(sin x,x=(1 .. 2));&    -1 .. 1\\
-on rounded;\\
-bounds(sin x,x=(1 .. 2));&    0.84147098481 .. 1\\
-bounds(x**2+x,x=(-0.5 .. 0.5));& - 0.25 .. 0.75\\
-\end{Examples}
-\end{Operator} 
-    
-\begin{Concept}{Chebyshev fit}
-\index{approximation}
-The operator family \name{Chebyshev\_...} implements approximation
-and evaluation of functions by the Chebyshev method.
-Let \name{T(n,a,b,x)} be the Chebyshev polynomial of order \name{n}
-transformed to the interval \name{(a,b)}. 
-Then a function \name{f(x)} can be
-approximated in \name{(a,b)} by a series
-
-\begin{verbatim}
-  for i := 0:n sum c(i)*T(i,a,b,x)
-\end{verbatim}
-
-The operator \name{chebyshev\_fit} computes this approximation and
-returns a list, which has as first element the sum expressed
-as a polynomial and as second element the sequence
-of Chebyshev coefficients.
-\name{Chebyshev\_df} and \name{Chebyshev\_int} transform a Chebyshev
-coefficient list into the coefficients of the corresponding
-derivative or integral respectively. For evaluating a Chebyshev
-approximation at a given point in the basic interval the
-operator \name{Chebyshev\_eval} can be used.  
-\name{Chebyshev\_eval} is based on a recurrence relation which is
-in general more stable than a direct evaluation of the 
-complete polynomial.
-
-\begin{Syntax}
-   \name{chebyshev\_fit}\(\meta{fcn},\meta{var}=\(\meta{lo} .. \meta{hi}\),\meta{n}\)
-
-   \name{chebyshev\_eval}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\),
-                            \meta{var}=\meta{pt}\)
-
-   \name{chebyshev\_df}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\)\)
-
-   \name{chebyshev\_int}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\)\)
-\end{Syntax}
-
-where \meta{fcn} is an algebraic expression (the target function),
-\meta{var} is the variable of \meta{fcn}, 
-\meta{lo} and \meta{hi} are
-numerical real values which describe an \nameref{interval} \meta{lo} < \meta{hi},
-the integer \meta{n} is the approximation order (set to 20 if missing),
-\meta{pt} is a number in the interval and \meta{coeffs} is
-a series of Chebyshev coefficients.
-
-\begin{Examples}
-
-on rounded;\\
-
-w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);&
-\begin{multilineoutput}{6cm}
-w := \{0.03824*x^3  - 0.2398*x^2  + 0.06514*x + 0.9778,
-      \{0.8991,-0.4066,-0.005198,0.009464,-0.00009511\}\}                    
-\end{multilineoutput}\\
-chebyshev_eval(second w, x=(1 .. 3), x=2.1);& 0.4111\\
-\end{Examples}
-\end{Concept}
-
-\begin{Operator}{num_fit}
-\index{approximation}\index{least squares}
-The operator \name{num_fit} finds for a set of
-points the linear combination of a given set of
-functions (function basis) which approximates the
-points best under the objective of the \name{least squares}
-criterion (minimum of the sum of the squares of the deviation).
-The solution is found as zero of the
-gradient vector of the sum of squared errors.
-
-
-\begin{Syntax}
-   \name{num_fit}\(\meta{vals},\meta{basis},\meta{var}=\meta{pts}\)
-\end{Syntax}
- 
-where \meta{vals} is a list of numeric values,
-\meta{var} is a variable used for the approximation,
-\meta{pts} is a list of coordinate values which correspond to 
-\meta{var},
-\meta{basis} is a set of functions varying in \name{var} which is used
-  for the approximation.
- 
-
-The result is a list containing as first element the
-function which approximates the given values, and as
-second element a list of coefficients which were used
-to build this function from the basis.
-  
-\begin{Examples}
- 
-pts:=for i:=1 step 1 until 5 collect i$\\
-vals:=for i:=1 step 1 until 5 collect\\
-            for j:=1:i product j$\\
-num_fit(vals,{1,x,x**2},x=pts);&
-\begin{multilineoutput}{6cm}
-    \{14.571428571*X^2  - 61.428571429*X + 54.6,\{54.6,
-         - 61.428571429,14.571428571\}\}
-\end{multilineoutput}
-
-\end{Examples}
-
-\end{Operator}
+\section{Numeric Package}
+\begin{Introduction}{Numeric Package}
+The numeric package supplies algorithms based on approximation
+techniques of numerical mathematics. The algorithms use
+the \nameref{rounded} mode arithmetic of REDUCE, including
+the variable precision feature which is exploited in some
+algorithms in an adaptive manner in order to reach the
+desired accuracy.
+\end{Introduction}
+
+\begin{Type}{Interval} 
+Intervals are generally coded as lower bound and
+upper bound connected by the operator \name{..}, usually 
+associated to a variable in an 
+equation. 
+
+\begin{Syntax}
+  \meta{var} = \(\meta{low} .. \meta{high}\)
+\end{Syntax}
+
+where \meta{var} is a \nameref{kernel} and \meta{low}, \meta{high} are
+numbers or expression which evaluate to numbers with \meta{low}<=\meta{high}.
+
+\begin{Examples}
+     x= (2.5 .. 3.5) 
+\end{Examples}
+
+means that the variable x is taken in the range from 2.5 up to
+3.5. 
+
+
+\end{Type}
+
+\begin{Concept}{numeric accuracy} 
+The keyword parameters \name{accuracy=a} and \name{iterations=i}, 
+where \name{a}and \name{i} must be positive integer numbers, control the
+iterative algorithms: the iteration is continued until
+the local error is below 10**{-a}; if that is impossible
+within \name{i} steps, the iteration is terminated with an
+error message. The values reached so far are then returned
+as the result.
+\end{Concept}
+
+\begin{Switch}{TRNUMERIC}
+Normally the algorithms produce only a minimum of printed
+output during their operation. In cases of an unsuccessful 
+or unexpected long operation a \name{trace of the iteration} can be
+printed by setting \name{trnumeric} \name{on}.
+\end{Switch}
+
+\begin{Operator}{num_min}
+\index{minimum}\index{steepest descent}\index{Fletcher Reeves} 
+The Fletcher Reeves version of the \name{steepest descent}
+algorithms is used to find the \name{minimum} of a
+function of one or more variables. The
+function must have continuous partial derivatives with respect to all
+variables. The starting point of the search can be
+specified; if not, random values are taken instead.
+The steepest descent algorithms in general find only local
+minima.
+ 
+\begin{Syntax}
+
+ \name{num\_min}\(\meta{exp}, 
+     \meta{var}[=\meta{val}] [,\meta{var}[=\meta{val}] ...
+             [,accuracy=\meta{a}] [,iterations=\meta{i}]\) 
+
+
+or
+
+ \name{num\_min}\(exp, \{ 
+      \meta{var}[=\meta{val}] [,\meta{var}[=\meta{val}] ...] \}
+             [,accuracy=\meta{a}] [,iterations=\meta{i}]\) 
+\end{Syntax}
+
+where \meta{exp} is a function expression,
+\meta{var} are the variables in \meta{exp} and
+\meta{val} are the (optional) start values.
+For \meta{a} and \meta{i} see \nameref{numeric accuracy}.
+
+\name{Num\_min} tries to find the next local minimum along the descending
+path starting at the given point. The result is a \nameref{list}
+with the minimum function value as first element followed by a list
+of \nameref{equation}\name{s}, where the variables are equated to the coordinates 
+of the result point.
+
+
+\begin{Examples}
+num_min(sin(x)+x/5, x)&\{4.9489585606,\{X=29.643767785\}\}\\
+num_min(sin(x)+x/5, x=0)&\{ - 1.3342267466,\{X= - 1.7721582671\}\}
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{num_solve}
+\index{equation solving}\index{equation system}\index{Newton iteration}
+\index{root}
+An adaptively damped Newton iteration is used to find
+an approximative root of a function (function vector) or the 
+solution of an \nameref{equation} (equation system). The expressions
+must have continuous derivatives for all variables.
+A starting point for the iteration can be given. If not given
+random values are taken instead. When the number of
+forms is not equal to the number of variables, the
+Newton method cannot be applied. Then the minimum
+of the sum of absolute squares is located instead. 
+
+With \nameref{complex} on, solutions with imaginary parts can be
+found, if either the expression(s) or the starting point
+contain a nonzero imaginary part. 
+
+\begin{Syntax}
+ \name{num\_solve}\(\meta{exp}, \meta{var}[=\meta{val}][,accuracy=\meta{a}][,iterations=\meta{i}]\)
+
+or
+
+ \name{num\_solve}\(\{\meta{exp},...,\meta{exp}\}, \meta{var}[=\meta{val}],...,\meta{var}[=\meta{val}]
+                [,accuracy=\meta{a}][,iterations=\meta{i}]\)
+
+or
+
+ \name{num\_solve}\(\{\meta{exp},...,\meta{exp}\}, \{\meta{var}[=\meta{val}],...,\meta{var}[=\meta{val}]\}
+                [,accuracy=\meta{a}][,iterations=\meta{i}]\)
+ 
+\end{Syntax} 
+
+
+where \meta{exp} are function expressions,
+      \meta{var} are the variables,
+      \meta{val} are optional start values.
+For \meta{a} and \meta{i} see \nameref{numeric accuracy}.
+ 
+\name{num_solve} tries to find a zero/solution of the expression(s).
+Result is a list of equations, where the variables are
+equated to the coordinates of the result point.
+ 
+The \nameindex{Jacobian matrix} is stored as side effect the shared
+variable \name{jacobian}.
+
+ 
+\begin{Examples}
+num_solve({sin x=cos y, x + y = 1},{x=1,y=2});&
+   \{X= - 1.8561957251,Y=2.856195584\}\\
+jacobian;&
+\begin{multilineoutput}{5cm}
+    [COS(X)  SIN(Y)]
+    [              ]
+    [  1       1   ]
+\end{multilineoutput}
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{num_int}
+\index{integration} 
+For the numerical evaluation of univariate integrals 
+over a finite interval the following strategy is used:
+If \nameref{int} finds a formal antiderivative
+    which is bounded in the integration interval, this
+    is evaluated and the end points and the difference
+    is returned.
+Otherwise a \nameref{Chebyshev fit} is computed, 
+    starting with order 20, eventually up to order 80.
+    If that is recognized as sufficiently convergent
+    it is used for computing the integral by directly
+    integrating the coefficient sequence.
+If none of these methods is successful, an
+    adaptive multilevel quadrature algorithm is used.
+
+For multivariate integrals only the adaptive quadrature is used.
+This algorithm tolerates isolated singularities. 
+The value \name{iterations} here limits the number of
+local interval intersection levels. 
+\meta{a} is a measure for the relative total discretization
+error (comparison of order 1 and order 2 approximations).
+
+ 
+\begin{Syntax}
+ \name{num_int}\(\meta{exp},\meta{var}=\(\meta{l} .. \meta{u}\)
+              [,\meta{var}=\(\meta{l} .. \meta{u}\),...]
+             [,accuracy=\meta{a}][,iterations=\meta{i}]\)
+\end{Syntax}
+
+where \meta{exp} is the function to be integrated,
+\meta{var} are the integration variables,
+\meta{l} are the lower bounds,
+\meta{u} are the upper bounds.
+ 
+Result is the value of the integral.
+ 
+\begin{Examples}
+num_int(sin x,x=(0 .. 3.1415926));& 2.0000010334
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{num_odesolve}
+\index{Runge-Kutta}\index{initial value problem}\index{ODE}
+The \name{Runge-Kutta} method of order 3 finds an approximate graph for
+the solution of real \name{ODE initial value problem}. 
+ 
+\begin{Syntax}
+ \name{num_odesolve}\(\meta{exp},\meta{depvar}=\meta{start},
+               \meta{indep}=\(\meta{from} .. \meta{to}\)
+                   [,accuracy=\meta{a}][,iterations=\meta{i}]\)
+
+or
+
+\name{num_odesolve}\(\{\meta{exp},\meta{exp},...\},
+                     \{ \meta{depvar}=\meta{start},\meta{depvar}=\meta{start},...\}
+               \meta{indep}=\(\meta{from} .. \meta{to}\)
+                   [,accuracy=\meta{a}][,iterations=\meta{i}]\)
+
+\end{Syntax}
+
+where 
+\meta{depvar} and \meta{start} specify the dependent variable(s)
+and the starting point value (vector), 
+\meta{indep}, \meta{from} and \meta{to} specify the independent variable
+and the integration interval (starting point and end point), 
+\meta{exp} are equations or expressions which
+contain the first derivative of the independent variable
+with respect to the dependent variable.
+ 
+The ODEs are converted to an explicit form, which then is
+used for a Runge Kutta iteration over the given range. The 
+number of steps is controlled by the value of \meta{i}
+(default: 20).  If the steps are too coarse to reach the desired 
+accuracy in the neighborhood of the starting point, the number is
+increased automatically.
+ 
+Result is a list of pairs, each representing a point of the
+approximate solution of the ODE problem.
+
+\begin{Examples}
+depend(y,x);\\
+num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5);&
+\begin{multilineoutput} 
+ \{\{0.0,1.0\},\{0.2,1.2214\},\{0.4,1.49181796\},\{0.6,1.8221064563\},
+  \{0.8,2.2255208258\},\{1.0,2.7182511366\}\}\\
+\end{multilineoutput}
+\end{Examples}
+
+In most cases you must declare the dependency relation
+between the variables explicitly using \nameref{depend};
+otherwise the formal derivative might be converted to zero.
+
+The operator \nameref{solve} is used to convert the form into
+an explicit ODE. If that process fails or if it has no unique result,
+the evaluation is stopped with an error message.
+
+\end{Operator}
+
+\begin{Operator}{bounds}
+
+Upper and lower bounds of a real valued function over an
+\nameref{interval} or a rectangular multivariate domain are computed
+by the operator \name{bounds}. The algorithmic basis is the computation
+with inequalities: starting from the interval(s) of the
+variables, the bounds are propagated in the expression
+using the rules for inequality computation. Some knowledge
+about the behavior of special functions like ABS, SIN, COS, EXP, LOG,
+fractional exponentials etc. is integrated and can be evaluated
+if the operator \name{bounds} is called with rounded mode on 
+(otherwise only algebraic evaluation rules are available).
+ 
+If \name{bounds} finds a singularity within an interval, the evaluation
+is stopped with an error message indicating the problem part
+of the expression.
+ 
+\begin{Syntax}
+  \name{bounds}\(\meta{exp},\meta{var}=\(\meta{l} .. \meta{u}\) 
+                 [,\meta{var}=\(\meta{l} .. \meta{u}\) ...]\)
+
+or
+
+  \name{bounds}\(\meta{exp},\{\meta{var}=\(\meta{l} .. \meta{u}\) 
+                 [,\meta{var}=\(\meta{l} .. \meta{u}\) ...]\}\)
+
+ 
+\end{Syntax}
+
+where \meta{exp} is the function to be investigated,
+\meta{var} are the variables of \meta{exp},
+\meta{l}  and  \meta{u} specify the area as set of \nameref{interval}\name{s}.
+
+\name{bounds} computes upper and lower bounds for the expression in the
+given area. An \nameref{interval} is returned.
+ 
+\begin{Examples}
+bounds(sin x,x=(1 .. 2));&    -1 .. 1\\
+on rounded;\\
+bounds(sin x,x=(1 .. 2));&    0.84147098481 .. 1\\
+bounds(x**2+x,x=(-0.5 .. 0.5));& - 0.25 .. 0.75\\
+\end{Examples}
+\end{Operator} 
+    
+\begin{Concept}{Chebyshev fit}
+\index{approximation}
+The operator family \name{Chebyshev\_...} implements approximation
+and evaluation of functions by the Chebyshev method.
+Let \name{T(n,a,b,x)} be the Chebyshev polynomial of order \name{n}
+transformed to the interval \name{(a,b)}. 
+Then a function \name{f(x)} can be
+approximated in \name{(a,b)} by a series
+
+\begin{verbatim}
+  for i := 0:n sum c(i)*T(i,a,b,x)
+\end{verbatim}
+
+The operator \name{chebyshev\_fit} computes this approximation and
+returns a list, which has as first element the sum expressed
+as a polynomial and as second element the sequence
+of Chebyshev coefficients.
+\name{Chebyshev\_df} and \name{Chebyshev\_int} transform a Chebyshev
+coefficient list into the coefficients of the corresponding
+derivative or integral respectively. For evaluating a Chebyshev
+approximation at a given point in the basic interval the
+operator \name{Chebyshev\_eval} can be used.  
+\name{Chebyshev\_eval} is based on a recurrence relation which is
+in general more stable than a direct evaluation of the 
+complete polynomial.
+
+\begin{Syntax}
+   \name{chebyshev\_fit}\(\meta{fcn},\meta{var}=\(\meta{lo} .. \meta{hi}\),\meta{n}\)
+
+   \name{chebyshev\_eval}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\),
+                            \meta{var}=\meta{pt}\)
+
+   \name{chebyshev\_df}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\)\)
+
+   \name{chebyshev\_int}\(\meta{coeffs},\meta{var}=\(\meta{lo} .. \meta{hi}\)\)
+\end{Syntax}
+
+where \meta{fcn} is an algebraic expression (the target function),
+\meta{var} is the variable of \meta{fcn}, 
+\meta{lo} and \meta{hi} are
+numerical real values which describe an \nameref{interval} \meta{lo} < \meta{hi},
+the integer \meta{n} is the approximation order (set to 20 if missing),
+\meta{pt} is a number in the interval and \meta{coeffs} is
+a series of Chebyshev coefficients.
+
+\begin{Examples}
+
+on rounded;\\
+
+w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);&
+\begin{multilineoutput}{6cm}
+w := \{0.03824*x^3  - 0.2398*x^2  + 0.06514*x + 0.9778,
+      \{0.8991,-0.4066,-0.005198,0.009464,-0.00009511\}\}                    
+\end{multilineoutput}\\
+chebyshev_eval(second w, x=(1 .. 3), x=2.1);& 0.4111\\
+\end{Examples}
+\end{Concept}
+
+\begin{Operator}{num_fit}
+\index{approximation}\index{least squares}
+The operator \name{num_fit} finds for a set of
+points the linear combination of a given set of
+functions (function basis) which approximates the
+points best under the objective of the \name{least squares}
+criterion (minimum of the sum of the squares of the deviation).
+The solution is found as zero of the
+gradient vector of the sum of squared errors.
+
+
+\begin{Syntax}
+   \name{num_fit}\(\meta{vals},\meta{basis},\meta{var}=\meta{pts}\)
+\end{Syntax}
+ 
+where \meta{vals} is a list of numeric values,
+\meta{var} is a variable used for the approximation,
+\meta{pts} is a list of coordinate values which correspond to 
+\meta{var},
+\meta{basis} is a set of functions varying in \name{var} which is used
+  for the approximation.
+ 
+
+The result is a list containing as first element the
+function which approximates the given values, and as
+second element a list of coefficients which were used
+to build this function from the basis.
+  
+\begin{Examples}
+ 
+pts:=for i:=1 step 1 until 5 collect i$\\
+vals:=for i:=1 step 1 until 5 collect\\
+            for j:=1:i product j$\\
+num_fit(vals,{1,x,x**2},x=pts);&
+\begin{multilineoutput}{6cm}
+    \{14.571428571*X^2  - 61.428571429*X + 54.6,\{54.6,
+         - 61.428571429,14.571428571\}\}
+\end{multilineoutput}
+
+\end{Examples}
+
+\end{Operator}